Abstract

This note concerns multiple weighted inequalities for vector-valued multilinear singular integral operator with nonsmooth kernel and its corresponding commutators containing multilinear
commutator and iterated commutator generated by the vector-valued multilinear operator and BMO functions. By the weighted estimates for a class of new variant maximal and sharp maximal functions, the multiple weighted norm inequalities for such operators are obtained.

1. Introduction

It is well known that multiple weighted norm inequalities for multilinear operators and their related commutators on various spaces of function is a center topic of harmonic analysis, which recently attracts a lot of attention, see [1–3] et al.

In this paper, we will focus on the multiple weighted estimates for vector valued multilinear singular integral with nonsmooth kernel and its commutators. Now we give some information on multilinear Calderón-Zygmund operators.

The multilinear operator we study is initially defined on the -fold product of Schwartz space and taking values into the space of tempered distributions ; that is,

A locally integrable function defined away from the diagonal in is called an associated kernel of if
where are functions with compact support and for all supp.

Moreover, we assume the associated kernel satisfies the following standard estimates:
for some and all with for some , and
for some and all , where . These facts can be founded in [4].

We now turn to present the definitions of -linear commutator and iterated commutator of multilinear singular integral.

For the multilinear operator and in , we define the -linear commutator as the following form:
where

If is associated with a distribution kernel, which coincides with the function defined away from the diagonal in , then
And we also present the iterated commutator ,

The following class of weights were introduced in [1]. Let , and , with , and ; given , we say that satisfies the condition if
where , when , is understood as .

Observe that is contained in for each . However, the class is not increasing with the natural partial order, see [1] for detail. Lerner et al. [1] established multiple weighted estimates for multilinear C-Z operators and that for -linear commutator of multilinear C-Z commutator. In 2012, Chen and Wu [2] extend their results to -linear commutator and iterated commutator of multilinear C-Z operator with nonsmooth kernel satisfying Assumptions (H1) and (H2).

Next we define the vector-valued multilinear operator associated with the operator by
where , for .

This operator was first studied by Grafakos and Martell in [7]. Later Cruz-Uribe et al. gained the weak boundedness of this one in [8]. We list them as follows.

Theorem A (see [7]). Let be a multilinear Calderón-Zygmund operator as before, and let , with , and with . There exists a constant such that

For the sequence , the vector-valued version of the commutators and are given by

In 2008, Tang established weighed norm inequalities for the commutators of vector-valued multilinear operator in [6], but their results are not the multiple weighted estimates that are obtained by Lerner et al. in [1].

Now we restore to give some information on the kernel which satisfy Assumptions (H1) and (H2). Let be a class of integral operators which play the role of the approximation to the identity. We always assume that the operators are associated with kernels in the sense that for all and
and that the kernels satisfy the following conditions
where is a positive fixed constant and is a positive, bounded, decreasing function satisfying that for some ,

Recall that the th transpose of the -linear operator is defined via
for all in . Notice that the kernel of is related to the kernel of via the identity

If an -linear operator maps a product of Banach Spaces to another Banach Space ; then transpose maps the product of Banach Spaces into . Moreover, the norms of and are equal. To maintain uniform notation, we may occasionally denote by and by .

Assumption (H1). Assume that for each , there exist operators with kernel satisfying conditions and and that for every , there exist kernels such that
for all in with . Also assume that for every and every , we have
where .

If satisfies Assumption (H1), we will say that is an -linear operator with generalized Caderón-Zygmund kernel . The collection of function satisfying (19) and (20) with parameters and will be denoted by , we say that is of class if has an associated kernel in .

Assumption (H2). Assume that there exist operators with kernel satisfying conditions (15) and (16) with constants and . Let
whenever and
whenever and .

When is of and its kernel also satisfies Assumption (H2), Duong et al. in [5] proved that multilinear singular integral is bounded from to for , with . And they also remarked that the above kernel which they studied has weaker regularity. It is natural to ask whether the vector-valued multilinear operator with kernels satisfying the same conditions as in [5] and its commutators and have multiple weighted estimates or not. These problems will be addressed by our next theorems.

Now we can formulate our results as follows.

Theorem 1. Assume that is a vector-valued multilinear operator defined as (10) associated with being an whose kernel satisfies Assumption (H2). If there exist , with and with then(i) can be extended to a bounded operator from to if all exponents are strictly greater than 1;(ii) can be extended to a bounded operator from to if , , and at least one of the .

Theorem 2. Assume that is a vector-valued multilinear operator that satisfies the assumptions in Theorem 1, and the multilinear commutator is defined as (12). Let , with , and with . Then there exists a constant such that

Theorem 3. Assume that is a vector-valued multilinear operator that satisfies the assumptions in Theorem 1, and the iterated commutator is defined as (13). Let , , with with . Then there exists a constant such that
where , , , and .

The rest of this paper is organized as follows. In Section 2, we recall some standard definitions and lemmas. In Section 3, we introduce a class of new maximal functions and prove some useful estimates which will play key roles in the proofs of our theorems. In Section 4, it is devoted to the proof our theorems. Throughout this paper, we use the letter to denote a positive constant that varies from line to line, but it is independent of the essential variable.

2. Some Preliminaries

Lemma 4 (see [1, Lemma 6.1]). Assume that satisfies condition, then there exists a finite constant such that .

We recall the definition of the Hardy-Littlewood maximal function and the sharp maximal function:
and their variants and .

We will use the following inequality (see [9]):
all functions for which the left-hand side is finite, and where , , .

Now we introduce some facts on Orlicz spaces. Let : be a Young function, that is, a continuous, convex, increasing function with and such that as . In this paper, any Young function will be doubling, namely for . We define the -average of function over a cube by
It is a simple but important observation that

A particular case of interest, and especially in this paper, is the Young function , the average of a function given by the Luxemburg norm .

Associated with this average, we have a maximal function
where the supremum is taken over all the cubes containing .

By the generalized Hölder’s inequality, we also get

3. New Maximal Functions

In this section, we will introduce certain variant multilinear maximal functions and establish the multiple weighted estimates for such functions, which are one of the main parts in this paper.

Recall the definitions of these maximal functions, which are introduced by Lerner et al. in [1]

The fact that , there exists a constant such that
so it is easy to check that

Characterizations of the multiple weights in terms of were proven in Theorems 3.3 and 3.7 in [1].

Lemma 5. Let , , and ,(i)If , then is bounded from to if and only if ;(ii)If , then is bounded from to if and only if .

In the following, we introduce the modified multilinear maximal functions.

Let , , and . We defined the following multilinear maximal functions:

where .

We remark that when and , was first introduced by Grafakos el al. in [10] and denoted by . Chen and Wu [2] proved the multiple weighted norm inequality for . Similarly to (33), for any , we have

Lemma 6. Let , , , , and , . Then for some ( depending only on ), and defined by (36) are bounded from to . , , , and are bounded from to .

4. Weighted Inequalities for Vector-Valued Singular Integral and Its Commutators

To prove our theorems, we first give two Lemmas about vector-valued operator associated with in which were obtained by Duong et al. in [5].

Lemma 7. Let be a multilinear operator in with satisfying Assumption (H2). And let , with , , and with . Then there exists a constant such that

Lemma 8. Let be a multilinear operator in with satisfying Assumption (H2). And let , with , and , with . Then there exists a constant such that

Before proving Theorem 1, we first present the estimates on the pointwise estimates for sharp Fefferman-stein maximal function action on .

Proposition 9. Let be an -linear operator in and satisfy the assumption in Theorem 1. Assume that and . For any in the product spaces with , for , then these exists a constant such that
The ideas and arguments used in the proof are similar to those in [1] with some modifications. For completeness, we give the proof as follows.

Proof of Proposition 9. For a point and a cube , since , for , to obtain (40), it suffices to prove for
for some constant to be determined later.Let be any smooth vector-valued functions. Set each , where and , then we can write
where each term of contains at least one .Hence, we can write
Applying Kolmogorov’s inequality to the term
with and , we derive
since .To estimate the remaining terms, we now set and will show that, for any ,
Consider the case when and define
So
Since , , and , then , , , and . Hence for . By Assumption (H2), we can infer
Thus,
It remains to estimate the terms in (46) with , for some for and , by Assumption (H2), we have
This finishes the proof of Proposition 9.

Proof of Theorem 1. By the definition of implies that . Using Proposition 9 and the Fefferman-Stein inequality (26) and observing that , we have
Then following from (52) and Lemma 6, Theorem 1 can be proved.We are left to check that is finite. The method of proof is similar to that of Theorem 3.19 in [2]. So we omit it here.

Proposition 10. Let be a vector-valued multilinear operator associated with an -linear operator in whose kernels satisfie the Assumption (H2). Suppose that is the corresponding commutator of with . Let , . Then there exists a constant depending on and , such that
holds for all bounded measurable vector functions .

Proof of Proposition 10. By linearity, it suffices to consider the operator with only symbols. Without loss of generality, we only consider the case: and denote by for convenience.Note that for any constant we have
Fix , for any cube centered at and a constant determined later, we have
We analyze each term separately. Recall that and and thanks to Hölder’s inequality and note that , it follows that
To estimate , we split again each into , where . This yields
where each term in contains at least one .Choose with , then
Noting that and and using Hölder’ inequality, we obtain
Now consider the term . By Hölder’s and Minkoswki’s inequalities as well as Assumptions (H1) and (H2), we obtain
Since , and , then , , and . Hence for .Thus from the Assumption (H2), we can follow that
where we use the fact that .What remains to be considered is the term such that , for some for and denote . We consider only the case , by Assumption (H2), we have